elliptic stochastic partial differential equations: an...

Elliptic stochastic partial differential equations:An orthonormal vector basis approach

S Adhikari1

1Swansea University, UK

Uncertainty Quantification Workshop, Edinburgh, 26 May, 2010

Adhikari (SU) Reduced methods for SPDE 26 May 2010 1 / 33

Outline of the talk

1 IntroductionStochastic elliptic PDEs

2 Spectral decomposition in a vector spaceProjection in a finite dimensional vector-spaceProperties of the spectral functions

3 Error minimization in the Hilbert spaceThe Galerkin approachComputational method

4 Numerical illustrationZnO nanowires

5 Conclusions


Introduction Stochastic elliptic PDEs

Stochastic elliptic PDE

We consider the stochastic elliptic partial differential equation(PDE)

−∇ [a(r, ω)∇u(r, ω)] = p(r); r in D (1)

with the associated boundary condition

u(r, ω) = 0; r on ∂D (2)

Here a : Rd × Ω→ R is a random field, which can be viewed as aset of random variables indexed by r ∈ Rd .We assume the random field a(r, ω) to be stationary and squareintegrable. Based on the physical problem the random field a(r, ω)can be used to model different physical quantities.



Discretized Stochastic PDE

The random process a(r, ω) can be expressed in a generalizedfourier type of series known as the Karhunen-Loeve expansion

a(r, ω) = a0(r) +∞∑

i=1

√νiξi(ω)ϕi(r) (3)

Here a0(r) is the mean function, ξi(ω) are uncorrelated standardGaussian random variables, νi and ϕi(r) are eigenvalues andeigenfunctions satisfying the integral equation∫D

Ca(r1, r2)ϕj(r1)dr1 = νjϕj(r2), ∀ j = 1,2, · · · .Truncating the series (3) upto the M-th term, substituting a(r, ω) inthe governing PDE (1) and applying the boundary conditions, thediscretized equation can be written as[

A0 +M∑

i=1

ξi(ω)Ai

]u(ω) = f (4)



Polynomial Chaos expansion

After the finite truncation, concisely, the polynomial chaosexpansion can be written as

u(ω) =P∑

k=1

Hk (ξ(ω))uk (5)

where Hk (ξ(ω)) are the polynomial chaoses.The value of the number of terms P depends on the number ofbasic random variables M and the order of the PC expansion r as

P =r∑

j=0

(M + j − 1)!

j!(M − 1)!(6)



Some basics of linear algebra

Definition

(Linearly independent vectors) A set of vectors φ1,φ2, . . . ,φn islinearly independent if the expression

∑nk=1 αkφk = 0 if and only if

αk = 0 for all k = 1,2, . . . ,n.

Remark

(The spanning property) Suppose φ1,φ2, . . . ,φn is a complete basisin the Hilbert space H. Then for every nonzero u ∈ H, it is possible tochoose α1, α2, . . . , αn 6= 0 uniquely such thatu = α1φ1 + α2φ2 + . . . αnφn.




We can ‘split’ the Polynomial Chaos type of expansions as

u(ω) =n∑

k=1

Hk (ξ(ω))uk +P∑

k=n+1

Hk (ξ(ω))uk (7)

According to the spanning property of a complete basis in Rn it isalways possible to project u(ω) in a finite dimensional vector basisfor any ω ∈ Ω. Therefore, in a vector polynomial chaos expansion(7), all uk for k > n must be linearly dependent.This is the motivation behind seeking a finite dimensionalexpansion.


Spectral decomposition in a vector space Projection in a finite dimensional vector-space

Projection in a finite dimensional vector-space

Theorem

There exist a finite set of functions Γk : (Rm × Ω)→ (R× Ω) and anorthonormal basis φk ∈ Rn for k = 1,2, . . . ,n such that the series

u(ω) =n∑

k=1

Γk (ξ(ω))φk (8)

converges to the exact solution of the discretized stochastic finiteelement equation (4) with probability 1.

Outline of the proof: The first step is to generate a completeorthonormal basis. We use the eigenvectors φk ∈ Rn of the matrix A0such that

A0φk = λ0kφk ; k = 1,2, . . .n (9)




Suppose the solution of Eq. (4) is given by

u(ω) =

[A0 +

M∑i=1

ξi(ω)Ai

]−1

f (10)

Using the eigenvector matrix and the orthonormality of Φ one has

u(ω) =

[Φ−TΛ0Φ

−1 +M∑

i=1

ξi(ω)Φ−T AiΦ−1

]−1

f = ΦΨ (ξ(ω))ΦT f

(11)where

Ψ (ξ(ω)) =

[Λ0 +

M∑i=1

ξi(ω)Ai

]−1

(12)

and the M-dimensional random vector

ξ(ω) = ξ1(ω), ξ2(ω), . . . , ξM(ω)T (13)




Now we separate the diagonal and off-diagonal terms of the Aimatrices as

Ai = Λi + ∆i , i = 1,2, . . . ,M (14)

Here the diagonal matrix

Λi = diag[A]

= diag[λi1 , λi2 , . . . , λin

]∈ Rn×n (15)

and ∆i = Ai − Λi is an off-diagonal only matrix.

Ψ (ξ(ω)) =

Λ0 +M∑

i=1

ξi(ω)Λi︸︷︷︸Λ(ξ(ω))

+M∑

i=1

ξi(ω)∆i︸︷︷︸∆(ξ(ω))

−1

(16)

where Λ (ξ(ω)) ∈ Rn×n is a diagonal matrix and ∆ (ξ(ω)) is anoff-diagonal only matrix.




We rewrite Eq. (16) as

Ψ (ξ(ω)) =[Λ (ξ(ω))

[In + Λ−1 (ξ(ω))∆ (ξ(ω))

]]−1(17)

The above expression can be represented using a Neumann type ofmatrix series as

Ψ (ξ(ω)) =∞∑

s=0

(−1)s[Λ−1 (ξ(ω))∆ (ξ(ω))

]sΛ−1 (ξ(ω)) (18)




Taking an arbitrary r -th element of u(ω), Eq. (11) can be rearranged tohave

ur (ω) =n∑

k=1

Φrk

n∑j=1

Ψkj (ξ(ω))(φT

j f) (19)

Defining

Γk (ξ(ω)) =n∑

j=1

Ψkj (ξ(ω))(φT

j f)

(20)

and collecting all the elements in Eq. (19) for r = 1,2, . . . ,n one has

u(ω) =n∑

k=1

Γk (ξ(ω))φk (21)


Spectral decomposition in a vector space Properties of the spectral functions

Spectral functions

DefinitionThe functions Γk (ξ(ω)) , k = 1,2, . . .n are called the spectral functionsas they are expressed in terms of the spectral properties of thecoefficient matrices of the governing discretized equation.

The main difficulty in applying this result is that each of thespectral functions Γk (ξ(ω)) contain infinite number of terms andthey are highly nonlinear functions of the random variables ξi(ω).For computational purposes, it is necessary to truncate the seriesafter certain number of terms.Different order of spectral functions can be obtained by usingtruncation in the expression of Γk (ξ(ω))



First-order spectral functions

Definition

The first-order spectral functions Γ(1)k (ξ(ω)), k = 1,2, . . . ,n are

obtained by retaining one term in the series (18).

Retaining one term in (18) we have

Ψ(1) (ξ(ω)) = Λ−1 (ξ(ω)) or Ψ(1)kj (ξ(ω)) =

δkj

λ0k +∑M

i=1 ξi(ω)λik(22)

Using the definition of the spectral function in Eq. (20), the first-orderspectral functions can be explicitly obtained as

Γ(1)k (ξ(ω)) =

n∑j=1

Ψ(1)kj (ξ(ω))

(φT

j f)

=φT

k fλ0k +

∑Mi=1 ξi(ω)λik

(23)

From this expression it is clear that Γ(1)k (ξ(ω)) are non-Gaussian

random variables even if ξi(ω) are Gaussian random variables.Adhikari (SU) Reduced methods for SPDE 26 May 2010 14 / 33


Second-order spectral functions

Definition

The second-order spectral functions Γ(2)k (ξ(ω)), k = 1,2, . . . ,n are

obtained by retaining two terms in the series (18).

Retaining two terms in (18) we have

Ψ(2) (ξ(ω)) = Λ−1 (ξ(ω))− Λ−1 (ξ(ω))∆ (ξ(ω))Λ−1 (ξ(ω)) (24)

Using the definition of the spectral function in Eq. (20), thesecond-order spectral functions can be obtained in closed-form as

Γ(2)k (ξ(ω)) =

φTk f

λ0k +∑M

i=1 ξi(ω)λik

−

n∑j=1

(φT

j f)∑M

i=1 ξi(ω)∆ikj(λ0k +

∑Mi=1 ξi(ω)λik

)(λ0j +

∑Mi=1 ξi(ω)λij

) (25)



Relationship with PC

Theorem

There exist a finite set of functions Γk : (Rm × Ω)→ (R× Ω) and anorthonormal basis φk ∈ Rn for k = 1,2, . . . ,n such that a vectorpolynomial chaos expansion can be expressed by

u(ω) =n∑

k=1

Γk (ξ(ω))φk (26)




Outline of the proof:

u(ω) = ui0h0 +∞∑

i1=1

ui1h1(ξi1(ω))

+∞∑

i1=1

i1∑i2=1

ui1,i2h2(ξi1(ω), ξi2(ω)) +∞∑

i1=1

i1∑i2=1

i2∑i3=1

ui1i2i3h3(ξi1(ω), ξi2(ω), ξi3(ω))

+∞∑

i1=1

i1∑i2=1

i2∑i3=1

i3∑i4=1

ui1i2i3i4 h4(ξi1(ω), ξi2(ω), ξi3(ω), ξi4(ω)) + . . . ,

(27)

where ui1,...,ip ∈ Rn are deterministic vectors to be determined. Usingthe spanning property of the orthonormal basis φk ∈ Rn in Remark 1,each of the ui1,...,ip can be uniquely expressed as

ui1,...,ip = α(1)i1,...,ip

φ1 + α(2)i1,...,ip

φ2 + . . .+ α(n)i1,...,ip

φn (28)Adhikari (SU) Reduced methods for SPDE 26 May 2010 17 / 33



Substituting this in Eq. (27) and collecting all the coefficientsassociated with each orthonormal vector φk the theorem is provedwhere

Γk (ξ(ω)) = α(k)i0

h0 +∞∑

i1=1

α(k)i1

h1(ξi1(ω))

+∞∑

i1=1

i1∑i2=1

α(k)i1,i2

h2(ξi1(ω), ξi2(ω)) +∞∑

i1=1

i1∑i2=1

i2∑i3=1

α(k)i1i2i3

h3(ξi1(ω), ξi2(ω), ξi3(ω))

+∞∑

i1=1

i1∑i2=1

i2∑i3=1

i3∑i4=1

α(k)i1i2i3i4

h4(ξi1(ω), ξi2(ω), ξi3(ω), ξi4(ω)) + . . . ,

(29)


Error minimization in the Hilbert space The Galerkin approach

The Galerkin approach

There exist a set of finite functions Γk : (Rm × Ω)→ (R× Ω), constantsck ∈ R and orthonormal vectors φk ∈ Rn for k = 1,2, . . . ,n such thatthe series

u(ω) =n∑

k=1

ck Γk (ξ(ω))φk (30)

converges to the exact solution of the discretized stochastic finiteelement equation (4) in the mean-square sense provided the vectorc = c1, c2, . . . , cnT satisfies the n × n algebraic equations S c = bwith

Sjk =M∑

i=0

Aijk Dijk ; ∀ j , k = 1,2, . . . ,n; Aijk = φTj Aiφk , (31)

Dijk = E[ξi(ω)Γj(ξ(ω))Γk (ξ(ω))

]and bj = E

[Γj(ξ(ω))

] (φT

j f).

(32)


Error minimization in the Hilbert space The Galerkin approach

The Galerkin approach

The error vector can be obtained as

ε(ω) =

(M∑

i=0

Aiξi(ω)

)(n∑

k=1

ck Γk (ξ(ω))φk

)− f ∈ Rn (33)

The solution is viewed as a projection where

Γk (ξ(ω))φk

∈ Rn

are the basis functions and ck are the unknown constants to bedetermined.We wish to obtain the coefficients ck such that the error normχ2 = 〈ε(ω), ε(ω)〉 is minimum. This can be achieved using theGalerkin approach so that the error is made orthogonal to thebasis functions, that is, mathematically

ε(ω)⊥(

Γj(ξ(ω))φj

)or

⟨Γj(ξ(ω))φj , ε(ω)

⟩= 0 ∀ j = 1,2, . . . ,n

(34)


Error minimization in the Hilbert space Computational method

Summary of the computational method

1 Solve the eigenvalue problem associated with the mean matrix A0to generate the orthonormal basis vectors: A0Φ = Λ0Φ

2 Select a number of samples, say Nsamp. Generate the samples ofbasic random variables ξi(ω), i = 1,2, . . . ,M.

3 Calculate the spectral basis functions (for example, first-order):

Γk (ξ(ω)) =φT

k fλ0k

+∑M

i=1 ξi (ω)λik

4 Obtain the coefficient vector: c = S−1b ∈ Rn, where b = f Γ,S = Λ0 D0 +

∑Mi=1 Ai Di and

Di = E[Γ(ω)ξi(ω)ΓT (ω)

], ∀ i = 0,1,2, . . . ,M

5 Obtain the samples of the response from the spectral series:u(ω) =

∑nk=1 ck Γk (ξ(ω))φk


Error minimization in the Hilbert space Computational method

Computational complexity

The spectral functions Γk (ξ(ω)) are highly non-Gaussian in natureand do not in general enjoy any orthogonality properties like theHermite polynomials or any other orthogonal polynomials withrespect to the underlying probability measure.The coefficient matrix S and the vector b should be obtainednumerically using the Monte Carlo simulation or other numericalintegration technique.The simulated spectral functions can also be ‘recycled’ to obtainthe statistics and probability density function (pdf) of the solution.The main computational cost of the proposed method depends on(a) the solution of the matrix eigenvalue problem, (b) thegeneration of the coefficient matrices Di , and (c) the calculation ofthe coefficient vector by solving linear matrix equation.For large M and n, asymptotically the computational costbecomes Cs = O(Mn2) + O(n3).


Numerical illustration ZnO nanowires

Collection of ZnO

Uncertainties in ZnO NWs in the close up view. The uncertainparameter include geometric parameters such as the length and thecross sectional area along the length, boundary condition and materialproperties.



ZnO nanowires

(a) The SEM image of a collection of ZnO NWshowing hexagonal cross sectional area.

(b) The continuum idealization of acantilevered ZnO NW under an AFMtip



Problem details

We study the deflection of ZnO NW under the AFM tip consideringstochastically varying bending modulus. The variability of thedeflection is particularly important as the harvested energy fromthe bending depends on it.We assume that the bending modulus of the ZnO NW is ahomogeneous stationary Gaussian random field of the form

EI(x , ω) = EI0(1 + a(x , ω)) (35)

where x is the coordinate along the length of ZnO NW, EI0 is theestimate of the mean bending modulus, a(x , ω) is a zero meanstationary Gaussian random field.The autocorrelation function of this random field is assumed to be

Ca(x1, x2) = σ2ae−(|x1−x2|)/µa (36)

where µa is the correlation length and σa is the standard deviation.



Problem details

We consider a long nanowire where the continuum model hasbeen validated.We use the baseline parameters for the ZnO NW from Gao andWang (Nano Letters 7 (8) (2007), 2499–2505) as the lengthL = 600nm, diameter d = 50nm and the lateral point force at thetip fT = 80nN.Using these data, the baseline deflection can be obtained asδ0 = 145nm. We normalize our results with this baseline value forconvenience.The correlation length considered in the numerical studies:µa = L/10.The number of terms M in the KL expansion becomes 67 (95%capture).The nanowire is divided into 50 beam elements of equal length.The number of degrees of freedom of the model n = 100(standard beam element).



Moments

(c) Mean of the normalized deflection. (d) Standard deviation of the normalizeddeflection.

Figure: The number of random variable used: M = 67. The number ofdegrees of freedom: n = 100.



Error in moments

Statistics Methods σa = 0.05 σa = 0.10 σa = 0.15 σa = 0.20Mean 1st order

Galerkin0.1761 0.7206 1.6829 3.1794

2nd orderGalerkin

0.0007 0.0113 0.0642 0.6738

Standard 1st orderGalerkin

3.9543 5.9581 9.0305 14.6568

deviation 2nd orderGalerkin

0.3222 1.8425 4.6781 8.9037

Percentage error in the mean and standard deviation of the deflectionof the ZnO NW under the AFM tip when correlation length is µa = L/3.For n = 100 and M = 67, if the second-order PC was used, one wouldneed to solve a linear system of equation of size 234,500. The resultsshown here are obtained by solving a linear system of equation of size100 using the proposed Galerkin approach.



Pdf

(a) Probability density function for σa =0.1.

(b) Probability density function for σa =0.2.

The probability density function of the normalized deflection δ/δ0 of theZnO NW under the AFM tip (δ0 = 145nm).


Conclusions

Conclusions

The only informaion used in the classical PC is the pdf of therandom variables.Here, additionally, the following information, coming from thediscreised PDE, are used:

A0 is symmetric and positive definite (used to generate theorthonormal basis)‖Ai‖ ≥ ‖Ai+1‖ , i = 0,1,2,3 . . . (used to generate the coefficientfunctions)

This is a ‘bespoke’ approach


Conclusions

Conclusions

(c) Basic building blocks. (d) Possible ‘solution’.

An analogy of PC based solution.Adhikari (SU) Reduced methods for SPDE 26 May 2010 31 / 33

Conclusions

Conclusions

Basic building blocks for the proposed method


Conclusions

Conclusions

1 We consider discretised stochastic elliptic partial differentialequations.

2 The solution is projected into a finite dimensional completeorthonormal vector basis and the associated coefficient functionsare obtained.

3 The coefficient functions, called as the spectral functions, areexpressed in terms of the spectral properties of the systemmatrices.

4 If n is the size of the discretized matrices and M is the number ofrandom variables, then the computational complexity grows inO(Mn2) + O(n3) for large M and n in the worse case.

5 We consider a problem with 67 random variables and n = 100degrees of freedom. A second-order PC would require thesolution of equations of dimension 234,500. In comparison, theproposed Galerkin approach requires the solution of algebraicequations of dimension n only.


elliptic stochastic partial differential equations: an...

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